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Math 105 Name : Exam 1 February 11, 2005
Show all your work. If you use your calculator to compute an answer, you must write down enough information on what you have done that your method is understandable.
(a) (5 pts.) State the formal limit definition of f ′(a).
(b) (9 pts.) Briefly explain the definition in part (a). For instance, what is it intended to calculate? What is the algebraic part of the expression calculating? Why is a limit necessary? You may include a graph as part of the explanation, but must verbally explain what the graph is indicating.
(c) (8 pts.) Use the definition in part (a) to show that if f (x) = x^2 − x, then f ′(3) = 5.
- (12 pts. – 6 pts. each) Give all possible correct answers to:
(a) If y = 2 cos x − x
3 3 + 3
√x, then dy dx =
(b) If g(t) =^2 t + 2t^ − 5 ln t, then g′(t) =
- (12 pts. – 6 pts. each) Give all possible correct answers to:
(a) If dT dr = r^3 +^2 r + 3 sin r, then T =
(b) If s′′(t) = −32, then s(t) =
- (6 pts. – 3 pts. each) Consider the differential equation dy dx =^ y^ −^ x. (a) Verify that y = Cex^ + x + 1 is a solution, for all values of C.
(b) Find the solution that satisfies the condition y(1) = 3.
- (9 pts. – 3 pts. each) Find the exact x-coordinates of the following on the graph of y = x^3 − 7 x + 5. Show enough work to justify your answers. (a) All local maxima.
(b) All local minima.
(c) All inflection points.
- (9 pts. – 3 pts. each) Give short answers (a sentence or two). (a) Can two different functions have the same rate of change? Explain.
(b) If a function f has an increasing derivative, then what does that say about the graph of f? Explain.
(c) If f has a stationary point at x = 2, must it have a local max or min there? Explain.
